Symplectic integration for optimal ergodic control

Autonomous active exploration requires search algorithms that can effectively balance the need for workspace coverage with energetic costs. We present a strategy for planning optimal search trajectories with respect to the distribution of expected information over a workspace. We formulate an iterative optimal control algorithm for general nonlinear dynamics, where the metric for information gain is the difference between the spatial distribution and the statistical representation of the time-averaged trajectory, i.e. ergodicity. Previous work has designed a continuous-time trajectory optimization algorithm. In this paper, we derive two discrete-time iterative trajectory optimization approaches, one based on standard first-order discretization and the other using symplectic integration. The discrete-time methods based on first-order discretization techniques are both faster than the continuous-time method in the studied examples. Moreover, we show that even for a simple system, the choice of discretization has a dramatic impact on the resulting control and state trajectories. While the standard discretization method turns unstable, the symplectic method, which is structure-preserving, achieves lower values for the objective.